TY - GEN
T1 - Estimation of Dependence in Multivariate Extreme Value Statistics
AU - Ho, Nguyen Khanh Le
PY - 2022/11/9
Y1 - 2022/11/9
N2 - In multivariate extreme value analysis, the extremal dependence structure between random variables can
be characterized in several ways. A complete characterization can be obtained from the spectral measure or
the stable tail dependence function. Alternatively, one can summarize the extremal dependency by a single
number, known as the tail dependence coefficient. This coefficient measures the strength of the extremal
dependence between the components of a bivariate random variable. Several well-performing estimators for
the tail dependence coefficient have been introduced in the literature. However, these estimators do not
take into account the presence of random covariates nor do they protect against potential contamination in
the given data, which will adversely affect the quality of the estimation, especially when relevant data are
scarce. To address these issues, we present a robust non-parametric method to estimate the conditional tail
dependence coefficient using the minimum density power divergence criterion. The asymptotic properties
of the estimator are studied under suitable regularity conditions.In order to mitigate the impacts of extreme events, numerous risk measures have been introduced in the
literature. In the multivariate context, the interest may be in the risk associated with one random variable
when a related variable becomes extreme. The marginal expected shortfall was introduced to quantify such
risk. Existing estimators for this measure do not take into account the presence of random covariates, which
can be incorporated to improve the accuracy of the estimation. As such, we present an estimator for the
conditional marginal expected shortfall. There are two main ingredients for this estimator: an in-sample
estimator and an extrapolation method when the related variable is extreme. These are studied separately
in two chapters, where the asymptotic normality of the proposed estimator in each chapter is established
for a wide class of conditional bivariate distributions, with heavy-tailed conditional marginal distributions
using empirical process arguments.Among existing risk measures in the literature is the marginal mean excess, which is the expected excess
of one risk above a high threshold conditional on a related variable exceeding another high threshold. In
this thesis, we introduce a generalization of this measure in the regression setting, the so-called conditional
marginal excess moment. We present an estimator for this new measure and establish its asymptotic normality under suitable conditions.The performance of the estimator presented in each chapter will be evaluated by a simulation study. The
efficiency and practical applicability will be illustrated on real datasets.
AB - In multivariate extreme value analysis, the extremal dependence structure between random variables can
be characterized in several ways. A complete characterization can be obtained from the spectral measure or
the stable tail dependence function. Alternatively, one can summarize the extremal dependency by a single
number, known as the tail dependence coefficient. This coefficient measures the strength of the extremal
dependence between the components of a bivariate random variable. Several well-performing estimators for
the tail dependence coefficient have been introduced in the literature. However, these estimators do not
take into account the presence of random covariates nor do they protect against potential contamination in
the given data, which will adversely affect the quality of the estimation, especially when relevant data are
scarce. To address these issues, we present a robust non-parametric method to estimate the conditional tail
dependence coefficient using the minimum density power divergence criterion. The asymptotic properties
of the estimator are studied under suitable regularity conditions.In order to mitigate the impacts of extreme events, numerous risk measures have been introduced in the
literature. In the multivariate context, the interest may be in the risk associated with one random variable
when a related variable becomes extreme. The marginal expected shortfall was introduced to quantify such
risk. Existing estimators for this measure do not take into account the presence of random covariates, which
can be incorporated to improve the accuracy of the estimation. As such, we present an estimator for the
conditional marginal expected shortfall. There are two main ingredients for this estimator: an in-sample
estimator and an extrapolation method when the related variable is extreme. These are studied separately
in two chapters, where the asymptotic normality of the proposed estimator in each chapter is established
for a wide class of conditional bivariate distributions, with heavy-tailed conditional marginal distributions
using empirical process arguments.Among existing risk measures in the literature is the marginal mean excess, which is the expected excess
of one risk above a high threshold conditional on a related variable exceeding another high threshold. In
this thesis, we introduce a generalization of this measure in the regression setting, the so-called conditional
marginal excess moment. We present an estimator for this new measure and establish its asymptotic normality under suitable conditions.The performance of the estimator presented in each chapter will be evaluated by a simulation study. The
efficiency and practical applicability will be illustrated on real datasets.
U2 - 10.21996/2h9f-js34
DO - 10.21996/2h9f-js34
M3 - Ph.D. thesis
PB - Syddansk Universitet. Det Naturvidenskabelige Fakultet
ER -